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We study the three point genus zero Gromov-Witten invariants on the Grassmannians which parametrize non-maximal isotropic subspaces in a vector space equipped with a nondegenerate symmetric or skew-symmetric form. We establish Pieri rules for the classical cohomology and the small quantum cohomology ring of these varieties, which give a combinatorial formula for the product of any Schubert class with certain special Schubert classes. We also give presentations of these rings, with integer coefficients, in terms of special Schubert class generators and relations.Comment: 59 pages, LaTeX, 6 figure

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We show that (equivariant) K-theoretic 3-point Gromov-Witten invariants of genus zero on a Grassmann variety are equal to triple intersections computed in the (equivariant) K-theory of a two-step flag manifold, thus generalizing an earlier result of Buch, Kresch, and Tamvakis. In the process we show that the Gromov-Witten variety of curves passing through 3 general points is irreducible and rational. Our applications include Pieri and Giambelli formulas for the quantum K-theory ring of a Grassmannian, which determine the multiplication in this ring. We also compute the dual Schubert basis for this ring, and show that its structure constants satisfy S 3 -symmetry. Our formula for Gromov-Witten invariants can be partially generalized to cominuscule homogeneous spaces by using a construction of Chaput, Manivel, and Perrin. Rationality of Gromov-Witten varietiesLet X = G/P be a homogeneous space defined by a complex connected semisimple linear algebraic group G and a parabolic subgroup P . An N -pointed stable map (of genus zero) to X is a morphism of varieties f : C → X, where C is a tree of projective lines, together with N distinct non-singular marked points of C, ordered from 1 to N , such that any component of C that is mapped to a single point in X contains at least three special points, where special means marked or singular [36].The degree of f is the homology class f * [C] ∈ H 2 (X; Z).A fundamental tool in Gromov-Witten theory is Kontsevich's moduli space M 0,N (X, d), which parametrizes all N -pointed stable maps to X of degree d. This space is equipped with evaluation maps ev i : M 0,N (X, d) → X for 1 ≤ i ≤ N , where ev i sends a stable map f to its image of the i-th marked point in its domain C. We let ev = ev 1 × · · ·× ev N : M 0,N → X N := X × · · ·× X denote the total evaluation map. For N ≥ 3, there is also a forgetful map ρ : M 0,N (X, d) → M 0,N := M 0,N (point, 0) which sends a stable map to its domain (after collapsing unstable components). The (coarse) moduli space M 0,N (X, d) is a normal projective variety with at worst finite quotient singularities, and its dimension is given byWe refer to the notes [19] for the construction of this space. It has been proved by Kim and Pandharipande [31] and by Thomsen [53] that the Kontsevich space M 0,N (X, d) is irreducible. Kim and Pandharipande also showed that M 0,N (X, d) is rational.

Abstract. A previous result of the authors with Chaput and Perrin states that the closure of the union of all rational curves of fixed degree passing through a Schubert variety in a homogeneous space G/P is again a Schubert variety. In this paper we identify this Schubert variety explicitly in terms of the Hecke product of Weyl group elements. We apply our result to give an explicit formula for any two-point Gromov-Witten invariant as well as a new proof of the quantum Chevalley formula and its equivariant generalization. We also recover a formula for the minimal degree of a rational curve between two given points in a cominuscule variety.